c.1200, brymme "edge of the sea," of obscure origin, perhaps akin to Old Norse barmr "rim, brim," probably related to German bräme "margin, border, fringe," from PIE *bhrem- "point, spike, edge." (Old English had brim in the sense "sea, surf," but this probably was from the Germanic stem *brem- "to roar, rage.") Extended by 1520s to cups, basins, hats.

v.

"to fill to the brim," 1610s, from brim (n.). Intransitive sense ("be full to the brim") attested from 1818. Related: Brimmed; brimming.